- Series and parallel circuits
If two or more circuit components are connected end to end like a daisy chain, it is said they are connected in

**"series**". A series circuit provides a single path forelectric current through all of its components.If two or more circuit components are connected like the rungs of a ladder it is said they are connected in

**"parallel**". A parallel circuit provides separate paths for current through each of its components. A parallel circuit provides the samevoltage across all its components. An old term for devices connected in parallel is "multiple", such as a multiple connection forarc lamp s.As an example, consider a very simple circuit consisting of four light bulbs and one 6 V battery. If a wire joins the battery to one bulb, to the next bulb, to the next bulb, to the next bulb, then back to the battery, in one continuous loop, the bulbs are said to be in series. If each bulb is wired to the battery in a separate loop, the bulbs are said to be in parallel. If the four light bulbs are connected in series the same current flows through all of them; 1.5 V is applied across each light bulb. If the light bulbs are connected in parallel, the current flowing through the light bulbs combine to form the current flowing in the battery; 6 V is applied across each light bulb.

In a series circuit, every device must function. A set of light bulbs in series with one bulb burning out, kills the circuit. In parallel, every light has its own circuit, so all but one light could be burned out, and the last one will still illuminate.

**Series circuits**Series circuits are sometimes called "current"-coupled or

daisy chain -coupled. The current that flows in a series circuit will flow through every component in the circuit. Therefore, all of the components in a series connection carry the same current.**Resistors**To find the total resistance of all the components, add the individual resistances of each component:

:$R\_mathrm\{total\}\; =\; R\_1\; +\; R\_2\; +\; R\_3\; +\; cdots\; +\; R\_n$

for components in series with resistances $R\_1$, $R\_2$, etc. To find the current $I$, use

Ohm's law ::$I\; =\; frac\{V\}\{R\_mathrm\{total$.To find the

voltage across a component with resistance $R\_i$, use Ohm's law again::$V\_i\; =\; IR\_i\; ,$where $I$ is the current, as calculated above. The components divide the voltage according to their resistances, so, in the case of two resistors,:$frac\{V\_1\}\{V\_2\}\; =\; frac\{R\_1\}\{R\_2\}$.

**NOTE:**The aboveformulae extend toimpedances in series.**Inductors**Inductor s follow the same law, in that the totalinductance of non-coupled inductors in series is equal to the sum of their individual inductances::$L\_mathrm\{total\}\; =\; L\_1\; +\; L\_2\; +\; cdots\; +\; L\_n$

However, in some situations it is difficult to prevent adjacent inductors from influencing each other, as the magnetic field of one device couples with the windings of its neighbours. This influence is defined by the mutual inductance M. For example, if you have two inductors in series, there are two possible equivalent inductances depending on how the magnetic fields of both inductors influence each other.

When there are more than two inductors, the mutual inductance between each of them and the way the coils influence each other complicates the calculation. For a larger number of coils the total combined inductance is given by the sum of all mutual inductances between the various coils including the mutual inductance of each given coil with itself, which we term self-inductance or simply inductance. For three coils, there are six mutual inductances $M\_\{12\}$, $M\_\{13\}$, $M\_\{23\}$ and $M\_\{21\}$, $M\_\{31\}$ and $M\_\{32\}$. There are also the three self-inductances of the three coils: $M\_\{11\}$, $M\_\{22\}$ and $M\_\{33\}$.

Therefore:$L\_mathrm\{total\}\; =\; (M\_\{11\}\; +\; M\_\{22\}\; +\; M\_\{33\})\; +\; (M\_\{12\}\; +\; M\_\{13\}\; +\; M\_\{23\})\; +\; (M\_\{21\}\; +\; M\_\{31\}\; +\; M\_\{32\})$

By reciprocity $M\_\{ij\}$ = $M\_\{ji\}$ so that the last two groups can be combined. The first three terms represent the sum of the self-inductances of the various coils. The formula is easily extended to any number of series coils with mutual coupling. The method can be used to find the self-inductance of large coils of wire of any cross-sectional shape by computing the sum of the mutual inductance of each turn of wire in the coil with every other turn since in such a coil all turns are in series.

**Capacitors**Capacitor s follow the same law using the reciprocals. The totalcapacitance of capacitors in series is equal to the reciprocal of the sum of the reciprocals of their individual capacitances::$frac\{1\}\{C\_mathrm\{total\; =\; frac\{1\}\{C\_1\}\; +\; frac\{1\}\{C\_2\}\; +\; cdots\; +\; frac\{1\}\{C\_n\}$.

The working voltage of a series combination of identical capacitors is equal to the sum of voltage ratings of individual capacitors provided that equalizing resistors are used to ensure equal voltage division. This is all because of Ohm's law $V\; =\; RI$

**Memristors**Memristor s in series are given by the sum of their memristance: :$M\_mathrm\{total\}\; =\; M\_1\; +\; M\_2\; +\; cdots\; +\; M\_n$**witches**Two or more

switch es in series form a logical AND; the circuit only carries current if all switches are 'on'. SeeAND gate .**Parallel circuits**If two or more components are connected in parallel they have the same potential difference (

voltage ) across their ends. The potential differences across the components are the same in magnitude, and they also have identical polarities. Hence, the same voltage is applicable to all circuit components connected in parallel. The total current "I" is the sum of the currents through the individual components, in accordance withKirchhoff's circuit laws . The current in each individual resistor is found byOhm's law . Factoring out the voltage gives:$I\_mathrm\{total\}\; =\; Vleft(frac\{1\}\{R\_1\}\; +\; frac\{1\}\{R\_2\}\; +\; cdots\; +\; frac\{1\}\{R\_n\}\; ight)$.**Notation**The parallel property can be represented in equations by two vertical lines $parallel$ (as in geometry) to simplify the equations.

**Resistors**To find the total resistance of all components, add the

reciprocal s of the resistances $R\_i$ of each component and take the reciprocal of the sum::$frac\{1\}\{R\_mathrm\{total\; =\; frac\{1\}\{R\_1\}\; +\; frac\{1\}\{R\_2\}\; +\; cdots\; +\; frac\{1\}\{R\_n\}$.

To find the current in a component with resistance $R\_i$, use Ohm's law again::$I\_i\; =\; frac\{V\}\{R\_i\},$.

The components divide the current according to their reciprocal resistances, so, in the case of two resistors,:$frac\{I\_1\}\{I\_2\}\; =\; frac\{R\_2\}\{R\_1\}$.

**NOTE:**The aboveformulae extend toimpedances in parallel.**Inductors**Inductor s follow the same law, in that the totalinductance of non-coupled inductors in parallel is equal to the reciprocal of the sum of the reciprocals of their individual inductances::$frac\{1\}\{L\_mathrm\{total\; =\; frac\{1\}\{L\_1\}\; +\; frac\{1\}\{L\_2\}\; +\; cdots\; +\; frac\{1\}\{L\_n\}$.

If the inductors are situated in each other's magnetic fields, this approach is invalid due to mutual inductance. If the mutual inductance between two coils in parallel is M, the equivalent inductor is::$frac\{1\}\{L\_mathrm\{total\; =\; frac\{L\_1+L\_2-2M\}\{L\_1L\_2-M^2\; \}$

If $L\_1=L\_2$:$L\_\{total\}\; =\; frac\{L+M\}\{2\}$

The sign of $M$ depends on how the magnetic fields influence each other. For two equal tightly coupled coils the total inductance is close to that of each single coil. If the polarity of one coil is reversed so that M is negative, then the parallel inductance is nearly zero or the combination is almost non-inductive. We are assuming in the "tightly coupled" case M is very nearly equal to L. However, if the inductances are not equal and the coils are tightly coupled there can be near short circuit conditions and high circulating currents for both positive and negative values of M, which can cause problems.

More than 3 inductors becomes more complex and the mutual inductance of each inductor on each other inductor and their influence on each other must be considered. For three coils, there are three mutual inductances $M\_\{12\}$, $M\_\{13\}$ and $M\_\{23\}$. This is best handled by matrix methods and summing the terms of the inverse of the $L$ matrix (3 by 3 in this case).

The pertinent equations are of the form:$v\_\{i\}=sum\_\{j\}\; L\_\{i,j\}frac\{di\_\{j\{dt\}$

**Capacitors**Capacitor s follow the same law using the reciprocals. The totalcapacitance of capacitors in parallel is equal to the sum of their individual capacitances::$C\_mathrm\{total\}\; =\; C\_1\; +\; C\_2\; +\; cdots\; +\; C\_n$.

The working voltage of a parallel combination of capacitors is always limited by the smallest working voltage of an individual capacitor.

**Memristors**Memristor s in parallel are given by:$M\_\{total\}\; =\; M\_1\; |\; M\_2\; |\; cdots\; |\; M\_n\; =\; left(\; M\_1^\{-1\}\; +\; M\_2^\{-1\}\; +\; cdots\; +\; M\_n^\{-1\}\; ight)^\{-1\}$**witches**Two or more

switch es in parallel, form a logical OR; the circuit carries current if at least one switch is 'on'. SeeOR gate .**Applications**Series circuits were formerly used for lighting in

electric multiple unit trains. For example, if the supply voltage was 600 volts there might be eight 70 volt bulbs in series (total 560 volts) plus aresistor to drop the remaining 40 volts. Series circuits for train lighting were superseded, first bymotor-generator s, then by solid state devices.**See also***

Wheatstone bridge

*Y-Δ transform

*Voltage divider

*Current divider

*Combining impedances**External links*** [

*http://www.diracdelta.co.uk/science/source/r/e/resistance/source.html Calculators for resistors in series and parallel.*]

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2010.*

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